12:00 AM
You just need to know that a bounded sequence has a convergent subsequence, aka Bolzano-Weierstrass.
If you don't have that basic result, you're kind of screwed.

Does this work if we use an actual example so I can see better how you're using this notation?

@0celo7 so can you share some wisdom on how to deal with this?

@user525966 Consider $1-x^2$ on $[-1,1]$

okay

@Cows No idea, I don't know how to integrate

12:01 AM
Hey @TedShifrin

(max value will end up being at x=0)

Yes

just for my own reference

@0celo7 is there a clever alternative to integration that you can share?

Mathematica

12:02 AM
so we want to show that the following must be false: $\forall d \in [-1,1], \exists x \in [-1,1]: f(x) > f(d)$
correct?

Ugh, what?
Is there some reason why you're insisting on the set theory

@0celo7 don't laugh but I don't know how to do this integral in Mathematica. I tried and it hates either it or me lol. Can you try it, please? and if it works tell me the code you used

I defined maximum value as $\exists d \in [a,b] : \forall x \in [a,b], ~f(x) \leq f(d)$

I mean that’s a reasonable statement, I don’t see the set theory

12:05 AM
@AkivaWeinberger you mean in mathematica or for the project?

sup yall

@user525966 Ok that's not wrong, but writing this out in terms of logical quantifiers isn't helping.

@AkivaWeinberger integrate[Sqrt[-3 - 12 t + 3 t^2 - 6 t^3 + 2 t^4], t]

does anyone have a good reference for an introductory topological dynamics text?

Do you know Bolzano-Weierstrass/Heine-Borel or not?

12:06 AM
@0celo7 integrate[Sqrt[-3 - 12 t + 3 t^2 - 6 t^3 + 2 t^4], t]

My concern here is that I don't want to prove something obvious with something else "obvious", I want it to be a natural outcome
i want to know why we need to do x, y, z, etc to get the proof
I do not

Because it's a nontrivial result.

@AkivaWeinberger mathematica just gives me back what i typed in

Just like the intermediate value theorem. There's no reason for it to be easy to prove.

@AkivaWeinberger mathematica just give me back what i gave it

12:07 AM
@Cows Mathematica commands start with capital letters, so what you wrote cannot be correct.

for example the concept of supremum almost feels like cheating because it has maximum practically built into it, so I don't see why this lets us prove something when it's basically the same as our original statement

Anyone good with highest weight stuff with reps?
In particular, Lie algebras.

@user525966 what?

@user525966 What?

All of math is about cheating , if you can get away with doing it , then do it , whatever it takes to make mathematical progress

12:09 AM
@Cows For the project… What $\int\sqrt{2t^4-6t^3+3t^2-12t-3}dt$ the simplification of something? What led you to that integral?

@0celo7 what is this (-(1/4) + t/3) Sqrt[-3 - 12 t + 3 t^2 - 6 t^3 + 2 t^4] +
1/4 (-((42 (t -
Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
2])^2 (-EllipticF[
ArcSin[\[Sqrt](((t -
Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
1]) (Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
2] - Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 +
2 #1^4 &, 4]))/((t -
Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
2]) (Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
1] - Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 +
2 #1^4 &,
4])))], -(((Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 +

Look, @Cows, I don't care. Sorry.

@0celo7 wow hehe

You never gave a good reason for doing this integral.

Oh God @Cows
Well, the "Root" stuff refers to roots of the polynomial, I guess
Various "Elliptic" functions are special functions

12:10 AM
Like when we start talking about "bounds" or "maxes" these notions practically get around the proof by asserting the max exists or something, to me that doesn't feel like a proof

@0celo7 I am trying to calculate the error from a cubic polynomial approximation of part of a circle. Please don't tell anyone lol

feels more like kicking the can down the road to some other concept that just takes something a true as an axiom somewhere

You can't prove anything about the reals without rigorously defining what the reals ARE.

Complete rigor is nonsense anyway , see Godel

There are two popular ways to do with: Dedekind cuts and equivalence classes of Cauchy sequences. (One can prove that the two approaches are equivalent.)

12:12 AM
i know the peano axioms for natural numbers, I never got to reals yet

So you only need rigor that’s enough to make your bosses happy

@Zee Gödel just means we can't prove everything, but we still can prove a whole lot of stuff

@0celo7 I mentioned how it came up. So is there something better I could do to get this?
@0celo7 to get a much nicer error?

@AkivaWeinberger you can prove whatever you want with the right set of axioms so my statement holds

I can prove anything if you give me contradictory axioms
I don't see the relevance

12:14 AM
@AkivaWeinberger yes i see. .
@AkivaWeinberger can I make Mathematica give me this result in latex form?

@Zee I don't see how Godel means you cannot have complete rigour.

Ok Let’s try a little exercise

@0celo7 Can I make Mathematica output this as latex?

@Cows I don't know enough about Mathematica

Give me a trivial but rigoures math statement

12:16 AM
Do you really need the indefinite integral? If you want a definite integral you could just get it to give you a decimal expansion

See any statement in Principia Mathematica, @Zee

@Zee $\forall X,\exists Y:X\in Y$

Or anything written in Coq.

$\forall x\in\Bbb R,\exists y:x<y$

Can you write that in English , I still can’t get latex to work

12:17 AM
First one: for every (set) X, there exists a Y such that X is in Y
Second one: for every number x, there exists a y such that x<y

Well that’s a bad example since those statements are already not well defined

@user525966 the reals are defined by the very axiom you are alluding to.

@JoeShmo Well, not really

Why are they not well-defined?

What the heck is Y ?

12:19 AM
@AkivaWeinberger well, what axiom, and how so? :-)

Dummy symbol ?

One way to define the reals is to say they are the unique ordered field with the least upper bound property (I think that's all the conditions), but we don't know that such a thing exists until we construct one
@Zee A set!
And then we construct one using Dedekind cuts or Cauchy sequences or what have you

what does it mean for x to be in Y? As a subset ? As an element ?

Element

@Zee It has much more to do with social effects and the strong history and tradition of academic work on leftism

12:21 AM
Ok , I still don’t know whats meant by a set but go ahead and prove this to me

∀X,∃Y:X∈Y

@AkivaWeinberger so we need the least upper bound property. What's the issue?

@0celo7 you know. . . you were somewhat rude dude. . . next time you act that way I might have to respond in kind. I was just asking what I was looking at. you could say I don't know rather than you don't care. Not that I care if you care, but . . .like I said kinda rude dude

@JoeShmo I don't think you'd define the reals by the least upper bound property, I think you'd define them by whatever construction you use. I guess the point doesn't really matter though
@Zee It follows from the axiom of pairing in ZFC
The axiom of pairing says that $\forall x\forall y\exists z(x\in z\land y\in z)$; take $x=y=X$ and $z=Y$
Obviously you're gonna have to reduce stuff to axioms, and you can't reduce the axioms to anything further. I just don't see the connection to Gödel.

Well what godel tells me , is that we should not lay axioms and fit math to them but the other way around

12:25 AM
the long boi in relative homology is basically a snake?
@AkivaWeinberger

And complete rigor is complete nonsense as well , that is true by common sense not even logic
“Mathematical logic is the least rigorous of all the sciences”
That’s what Gromov said in a talk once

@AkivaWeinberger if i remember correctly from model theory, any real closed field is elementarily equivalent to the reals. I.e., the l.u.b. property defines the reals in a very real (no pun intended) sense.

@JoeShmo l.u.b. is not a first order property

For my example of a rigorous statement I’d probably do something in graph theory

I am attempting to use TexForm so I can see what Mathematica is outputting in a way I can understand

12:29 AM
@LeakyNun ohkay?

How do I make mathematica output in tex form so that I can run it in tex and see what is saying?

Eg the graph K_3,2 is not planar

I’ll make it simpler 1+1=2
That’s a completely empirical fact

@JoeShmo the fact you quoted concerns first order theories

so what more am i needing exactly to prove extreme value theorem
apparently this subsequence concept

12:30 AM
Maybe that’s too complicated example on my part. Hmm

I don't use Mathematica often, can anyone tell me how to convert Mathematica speak to latex?

You may say , oh , we can define the natural integers using blah blah blah but to even talk about that you already assumed the concept of quantity

@Zee math is non-empirical, so I have no idea what you are saying.

Lolol
What else is it ? You think angels come at night and whisper mathematical ideas into your ear ?

“The complete graph on n nodes has n(n-1)/2 edges.”

12:31 AM
@JoeShmo Yeah I guess you're right

@LeakyNun in so far that a real closed fields has the same first order proeprties as the reals.

right

so whats the point you were making?

@Zee you think there is a physical, measurable, experimentally-observable phenomenon on earth that corresponds to every mathematical truth?

Philosophically, this would be the place to talk about whether there’s such a thing as synthetic a priori knowledge

12:32 AM
@Zee Arright fine, at the lowest levels maybe we can get a little nonrigorous, but the stuff connecting the lowest levels to the higher levels should be rigorous

But ugh I just find that boring

no, that would make mathematics as rigoures as physics which isn’t the case since physics is more rigoures

At this point you have to be pulling my leg.

@Semiclassical how can I clean up mathematica output?

12:33 AM
@BalarkaSen so the long boi in relative homology is basically a snake?

*rigorous @Zee
@LeakyNun Yeah

@Semiclassical I just want to be able to see it in human form, or latex or somethings

oh thanks

@AkivaWeinberger even then it’s not that simple , it’s just people being self honest and honest with each other, that’s as close to real rigor as you can get

This is reaching Poe’s law territory, yeah

12:34 AM
Isn't it called the snake lemma?

long boi has to be Tor, I think

@Cows ?
whatever

@AkivaWeinberger yes?

Tor and Ext
Def.

"What are the fewest axioms required to prove all the stuff we already know are true" is a reasonable question to ask
The fewer the axioms the less likely there's a contradiction

12:36 AM
looking into this compactness concept, "A topological space is compact if every open covering has a finite sub-covering."

"all the stuff we already know are true" isn't clear, Akiva. Do you mean after already assuming a model?

Right, so the main concept there that you don't know yet is "open covering"

Because that is called reverse mathematics.

The fewest axioms would be the set of all mathematical facts we “know” to be true today
Unfortunately they are probably inconsistent

We don't know anything to be true in math.
It's all conditional.

12:37 AM
@0celo7 I was hurt, man! I am just trying to figure out what I am looking at and you made me feel irrelevant. If I knew what to do I would not be asking, and i don't have people and faculty at my disposal. what I am saying is that your response was hurtful, but anyways I just want to be able to look at what Mathematica is saying in a way that I can follow. Can i have tried to use TexForm to convert to latex and it is not working

"If we assume this logical system, then we can conclude these things"

That’s what “” are for

For my part I don’t think mathematical logic is nonsense or lacking in rigor. I just largely find it boring

"An open covering of a space X is a collection {Ui} of open sets with unionoveri Ui = X and this has a finite sub-covering if a finite number of the Ui's can be chosen which still cover X."

But that's the problem, @Zee, you don't have a single thing that you "know" to be true. Because in one system, things can be true, but than in another they can be false.

12:39 AM
You're saying we don't even know that "If [if A then B] and [A] then [B]"? @Zee

There is no mathematical statement that is universally true across all logical models.

That said, the development of mathematical logic is strongly linked to the development of stuff like computer science

Yes Akiva

huh?

@user525966 What huh

12:40 AM
So I can respect logic as a discipline for that alone

this open covering concept
they seem to involve open sets again
which i still don't know if I understand yet

@0celo7 i am not calculating anything exotic. I am just trying to calculate the error from a simple approximation a high school kid can do, in case you are thinking I am (pretending) to do some exotic string physics

what is a closed set for example

Logic never gave us anything , intuition and imagination gave us the best of math

e.g. S = {1, 5, 8} ?

12:40 AM
@user525966 Just start with Rudin Chapter 1 and go to Chapter 8.

is it easy to read or is it going to be full of undefined notation and jargon you'd only understand by going through a bunch of classes

have a look
I'm currently on chapter 8

The paragraph in the Preface beginning with "Experience" is interesting @orbit-stabilizer

@Zee your lack of knowledge concerning the field of logic is showing.

nothing wrong with logic

12:44 AM
@Akiva, I sort of appreciate the construction of the reals, but I don't care for it. I'm not interested in going that deep into foundational stuff.

One thing I always find interesting to note about logic: Godel was a Platonist

By logic , I don’t really mean the field of mathematical logic , I mean logic as the mental faculty

There exists a unique ordered field with the lub property up to an isomorphism is enough for me.

Even logicians don’t use logic to discover new truths

Woah, what tf is wrong with logic as a mental faculty
Define, "new truths"

12:45 AM
Though I usually find discussions about argumentation and reasoning to be tiring as well

@AkivaWeinberger Are S = {1, 5, 8} and B = [1, 6] examples of closed sets?

One with finite cardinality, one with infinite cardinality?

Because it’s ultimately an argument about argument

Are you saying modus ponens doesn't come from the logical faculty?

12:46 AM
Which by it’s very nature presupposes it’s own validity

it does

And that’s the essence of it all semi

and this is why engineering > math

An argument that would deny argument is not speakable

12:47 AM
what's modus ponens again? (p -> q) ^ (p) -> q?

Nor one that affirms it

@user yeah
oops, pinged all the users

Not sure I agree there

@user525966 Yes

why are you even debating logic

12:48 AM
@AkivaWeinberger to my earlier question?

planes fly, computers work - life is good

@user525966 Click on the arrow next to the comment to see what I was replying to

Semi well, let’s just leave it there ;)

I think you can have arguments in support of reason. But there’s no external authority you can point to in order to settle it

@Semiclassical Unpopular opinion: inductive logic in science is illogical

12:49 AM
Metaphysics is exhausting

Philosophy is exhausting

I am exhausted

Expecting the sun to rise might be insane
There's no reason for it to rise
None that we know of at least

That's why it's best to be Bayesian

That’s the argument of Hume about inductive knowledge

12:50 AM
Can someone help me reason this

@0celo7 depends on what one means by logic

because the logic involved in assessing that truth statement requires a lot more behind it than just saying "the sun will rise"

Metaphysics is the worst nonsense there is.

I think Mathematica is blindly doing some computations here

you need a lot of ifs/premises to make that claim

12:51 AM
If one is hoping to acquire truth as a mirror of the world, sure

if I were to solve this by had, I would factor the quartic

@AkivaWeinberger look at the mess we made

not necessarily

and do partial fractions

@Cows ! This isn't a chat room about math!

12:51 AM
this would enable me to have a cleaner solution

We're discussing philosophy!
:P

If one is just hoping to make progress on interesting problems, then inductive logic is s perfectly sound tactic

@Semiclassical The probability that there's a man who can flip a switch and prevent the sun from coming up is the same as the sun just coming up again

Only tactic

indeed, both are 0

12:52 AM

because the number of possibilities are infinite
therefore, the sun will not rise
?

Assuming an infinite universe

I think I'm doing this wrong

Assuming a universe.

Can someone look at this integral for just even 10 minutes

12:53 AM

Mathematica is not helping

I have an answer, and it means nothing

Integrate over the differential form, Cows
INTEGRATE IT GOOD

12:54 AM
Lol war room

why are closed sets closed under both finite and infinitely many intersections, but only closed under finitely many unions

Are there any integral transforms that might help?

Yes , try the Fourier Mukai transform

@user525966 because one can prove the first, and give a counterexample for the second.

12:55 AM

I can't do the integral, too stupid

This ain’t the 40s we aren’t your computers , use the one in front of you

@zee let me look up the transforms
@zee I have worked on this for a few days
the last thing I attempted to do was factor the quartic using some number theory tricks, but i did not succeed

what's the counterexample? if I have a closed set {1} and i union it with closed set {2} then {3} then {4}... to infinity, does that work?

Look up some of the Soviet books filled with formulas for integrals

12:57 AM
that created set {1, ..., infinity}

@Zee I am not sure how to implement this by hand or computer

That was a joke

@zee oh my god lolz
dude I am desperate man